2019
DOI: 10.1016/j.jco.2019.01.002
|View full text |Cite
|
Sign up to set email alerts
|

Random bit multilevel algorithms for stochastic differential equations

Abstract: We study the approximation of expectations E(f (X)) for solutions X of SDEs and functionals f : C([0, 1], R r ) → R by means of restricted Monte Carlo algorithms that may only use random bits instead of random numbers. We consider the worst case setting for functionals f from the Lipschitz class w.r.t. the supremum norm. We construct a random bit multilevel Euler algorithm and establish upper bounds for its error and cost. Furthermore, we derive matching lower bounds, up to a logarithmic factor, that are valid… Show more

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
4

Citation Types

0
24
0

Year Published

2019
2019
2022
2022

Publication Types

Select...
3
2
1

Relationship

0
6

Authors

Journals

citations
Cited by 10 publications
(24 citation statements)
references
References 22 publications
0
24
0
Order By: Relevance
“…[11]), and consequently a careful analysis of d 1 (δ un • , µ) as n → ∞ is indispensable. In computational mathematics, best uniform approximations also arise naturally in the form of restricted MC methods, and a basic question is how performance of the latter compares to general (non-restricted) MC methods, that is, how d r (δ un • , µ) compares to d r (δ •,n • , µ) as n → ∞; see, e.g., [14,15] and the many references therein. Restricted MC methods have recently found applications in "big-data" problems in Bayesian statistics [25] and the numerical solution of SDE, notably in mathematical finance [14,16,28].…”
Section: Introductionmentioning
confidence: 99%
“…[11]), and consequently a careful analysis of d 1 (δ un • , µ) as n → ∞ is indispensable. In computational mathematics, best uniform approximations also arise naturally in the form of restricted MC methods, and a basic question is how performance of the latter compares to general (non-restricted) MC methods, that is, how d r (δ un • , µ) compares to d r (δ •,n • , µ) as n → ∞; see, e.g., [14,15] and the many references therein. Restricted MC methods have recently found applications in "big-data" problems in Bayesian statistics [25] and the numerical solution of SDE, notably in mathematical finance [14,16,28].…”
Section: Introductionmentioning
confidence: 99%
“…As a consequence, the lower bounds on the number of random bits from [11] also hold in this setting. We also derive a lower bound on the number of needed bits for integration of Lipschitz functions over the Wiener space, complementing a result of [5].…”
mentioning
confidence: 82%
“…Restricted Monte Carlo algorithms were considered in [12,13,16,11,14,3,17,4,5,6]. Restriction usually means that the algorithm has access only to random bits or to random variables with finite range.…”
Section: Introductionmentioning
confidence: 99%
See 1 more Smart Citation
“…An interesting variant of (1.4)-(1.6) considers best uniform approximations of µ ∈ P, that is, best approximations of µ by ν ∈ P * n , subject to the additional requirement that nν({x}) is a (positive) integer for every x ∈ supp ν. Best uniform (or, more generally, best constrained) approximations have recently attracted considerable interest, not least in view of potential applications in stochastic processes and differential equations [7,8,16,17,36,37]; they may also be viewed as deterministic analogues of (random) empirical measures [6,9,14]. With δ un • denoting a best uniform d-approximation of µ, trivially d(µ, δ •,n • ) ≤ d(µ, δ un • ).…”
Section: Introductionmentioning
confidence: 99%